GCSE revision - surds

What are surds?

A surd is an expression that includes a square root. If you look in your textbook (why?) it will say that surds are roots that cannot be expressed as rational numbers. This means that most roots are surds:

sqrt{2}

,

sqrt{3}

,

sqrt{24}

, the list goes on. You can try and work them out on a calculator, but that will only give you an estimate, accurate to ten decimal places or so - it's not exact. Surds are all "irrational" numbers - that means they can't be expressed as fractions, so their decimal expansions go on forever with no real pattern. (Fractions like

frac{1}{3}

or

frac{1}{7}

go on forever too, but they start repeating themselves after a while.)

Manipulations of surds

There are two basic identities you need to know.

• sqrt{a} times sqrt{b} = sqrt{ab}

  . For example,

  sqrt{2} times sqrt{3} = sqrt{6}

  .

• frac{sqrt{a}}{sqrt{b}} = sqrt{frac{a}{b}}

  . For example,

  frac{sqrt{20}}{sqrt{5}} = sqrt{frac{20}{5}} = sqrt{4}

IMPORTANT: There are no simple identities for adding and subtracting surds - in most cases, something like

sqrt{2}+sqrt{5}

can't be simplified!

Simplifying surds

For example:

sqrt{72} = sqrt{36} times sqrt{2} = 6sqrt{2}

To simplify a surd

sqrt{x}

, you have to find the largest perfect square that divides

x

. Above, that was 36. You then separate the two to get something of the form

asqrt{b}

. Sometimes this isn't easy - think of something like

sqrt{968}

! If you can't immediately find the largest factor, then, it's a good idea to get rid of smaller factors to simplify the problem. For example:

sqrt{968} = 2sqrt{242} = 2 times 11sqrt{2} = 22sqrt{2}

In extreme cases, just factorise the whole number (like in GCSE) and look for repeated factors:

sqrt{187187} = sqrt{7 times 11 times 11 times 13 times 17} = 11sqrt{7 times 13 times 17} = 11sqrt{1547}

Hooray for calculators!

Rationalising the denominator

This is the tricky bit. When you're dealing with fractions, Edexcel hates it when you leave surds on the bottom - you have to "rationalise the denominator". For example:

frac{1}{sqrt{5}} = frac{1}{sqrt{5}} times frac{sqrt{5}}{sqrt{5}} = frac{sqrt{5}}{5}

See what we did there? We wanted to get rid of the surd from the denominator, so we multiplied top and bottom by the surd. That's basically it. Another example:

frac{1}{2sqrt{5}} = frac{1}{2sqrt{5}} times frac{sqrt{5}}{sqrt{5}} = frac{sqrt{5}}{2 times sqrt{5}^2} = frac{sqrt{5}}{10}

There are some fractions where this method won't work, though, because the denominator has more than one term in it. In that case, we have to use the difference of two squares. Take a look:

frac{1}{sqrt{2}+sqrt{5}} = frac{1}{sqrt{2}+sqrt{5}} times frac{sqrt{2}-sqrt{5}}{sqrt{2}-sqrt{5}} = frac{sqrt{2}-sqrt{5}}{sqrt{2}^2 - sqrt{2}sqrt{5} + sqrt{2}sqrt{5} - sqrt{5}^2} = -frac{sqrt{2}-sqrt{5}}{3}

The irrational bit cancels, making life easier for us all!